Warm-up:1) If a particle has a velocity function

defined by , find its acceleration function.

2) If a particle has an acceleration function defined by , what is its velocity function? Is there more

than one possibility?

2563)( 34 ttttv

54)( 3 xta

IntegrationSection 6.1 & 6.2

The Area Under a Curve / Indefinite Integrals

The Rectangle Method for Finding Areas

• Read over Example on pg. 351 and draw a conclusion about the Area under of curve as it relates to the number of rectangular sub-intervals.

The Rectangle Method for Finding Areas

• Read over Example on pg. 351 and draw a conclusion about the Area under of curve as it relates to the number of rectangular sub-intervals.

• When we become more comfortable with integration we will use integrals to more accurately find the area under a curve.

Anti-differentiation (Integration)

• The opposite of derivatives (anti-derivatives)

• Ex: You are given a velocity function and you want to find out what the position function is for the particle. How would you determine s(t)?

Anti-differentiation (Integration)

• The opposite of derivatives (anti-derivatives)

• Ex: You are given a velocity function and you want to find out what the position function is for the particle. How would you determine s(t)?

• Let’s assume ttv 2

Anti-differentiation (Integration)

• The opposite of derivatives (anti-derivatives)

• Ex: You are given a velocity function and you want to find out what the position function is for the particle. How would you determine s(t)?

• Let’s assume

ttv 2

2tts

Anti-differentiation (Integration)

• The opposite of derivatives (anti-derivatives)

• Ex: You are given a velocity function and you want to find out what the position function is for the particle. How would you determine s(t)?

• Let’s assume

• Could work?

ttv 2

2tts

32 tts

Anti-differentiation (Integration)

• The opposite of derivatives (anti-derivatives)

• Ex: You are given a velocity function and you want to find out what the position function is for the particle. How would you determine s(t)?

• Let’s assume

• Could work? How about ?

ttv 2

2tts

32 tts 52 tts

Indefinite Integrals• The process of finding anti-derivatives is

called Anti-Differentiation or Integration.

•

Indefinite Integrals• The process of finding anti-derivatives is

called Anti-Differentiation or Integration.

• )()]([ xfxFdx

d

Indefinite Integrals• The process of finding anti-derivatives is

called Anti-Differentiation or Integration.

• can be written as using Integral Notation,

)()]([ xfxFdx

d CxFdxxf )()(

Indefinite Integrals• The process of finding anti-derivatives is

called Anti-Differentiation or Integration.

• can be written as using Integral Notation, where the expression is called an Indefinite Integral,

)()]([ xfxFdx

d CxFdxxf )()(

dxxf )(

Indefinite Integrals• The process of finding anti-derivatives is

called Anti-Differentiation or Integration.

• can be written as using Integral Notation, where the expression is called an Indefinite Integral, the function f(x) is called the Integrand,

)()]([ xfxFdx

d CxFdxxf )()(

dxxf )(

Indefinite Integrals• The process of finding anti-derivatives is

called Anti-Differentiation or Integration.

• can be written as using Integral Notation, where the expression is called an Indefinite Integral, the function f(x) is called the Integrand, and the constant C is called the Constant of Integration.

)()]([ xfxFdx

d CxFdxxf )()(

dxxf )(

Properties of Integrals:

Properties of Integrals:• A constant Factor can be moved through

an Integral sign:

Properties of Integrals:• A constant Factor can be moved through

an Integral sign:dxxfcdxxcf )()(

Properties of Integrals:• A constant Factor can be moved through

an Integral sign:

• An anti-derivative of a sum is the sum of the anti-derivatives:

dxxfcdxxcf )()(

Properties of Integrals:• A constant Factor can be moved through

an Integral sign:

• An anti-derivative of a sum is the sum of the anti-derivatives:

dxxfcdxxcf )()(

dxxgdxxfdxxgxf )()()]()([

Properties of Integrals:• A constant Factor can be moved through

an Integral sign:

• An anti-derivative of a sum is the sum of the anti-derivatives:

• An anti-derivative of a difference is the difference of the anti-derivatives:

dxxfcdxxcf )()(

dxxgdxxfdxxgxf )()()]()([

Properties of Integrals:• A constant Factor can be moved through

an Integral sign:

• An anti-derivative of a sum is the sum of the anti-derivatives:

• An anti-derivative of a difference is the difference of the anti-derivatives:

dxxfcdxxcf )()(

dxxgdxxfdxxgxf )()()]()([

dxxgdxxfdxxgxf )()()]()([

Integral Power Rule

• To integrate a power function (other than -1), add 1 to the exponent and divide by the new exponent.

Cr

xdxx

rr

1

1

Integral Power Rule

• To integrate a power function (other than -1), add 1 to the exponent and divide by the new exponent.

• Find

Cr

xdxx

rr

1

1

dxx23

Integral Power Rule

• To integrate a power function (other than -1), add 1 to the exponent and divide by the new exponent.

• Find

Cr

xdxx

rr

1

1

dxxdxx 22 33

Integral Power Rule

• To integrate a power function (other than -1), add 1 to the exponent and divide by the new exponent.

• Find

Cr

xdxx

rr

1

1

dxxdxx 22 33

Cx

)3(3

3

Integral Power Rule

• To integrate a power function (other than -1), add 1 to the exponent and divide by the new exponent.

• Find

Cr

xdxx

rr

1

1

dxxdxx 22 33

Cx

)3(3

3

Cx 3

Examples (S)1) Find

2) Find

3) Find

dxx2

dxx41

dxx

Examples (S)1) Find

2) Find

3) Find

Cx

dxx 3

32

dxx41

dxx

Examples1) Find

2) Find

3) Find

Cx

dxx 3

32

dxxdxx 44

1

dxx

Examples1) Find

2) Find

3) Find

Cx

dxx 3

32

Cx

dxxdxx

3

1 34

4

dxx

Examples1) Find

2) Find

3) Find

Cx

dxx 3

32

Cx

Cx

dxxdxx

3

34

4 3

1

3

1

dxx

Examples1) Find

2) Find

3) Find

Cx

dxx 3

32

Cx

Cx

dxxdxx

3

34

4 3

1

3

1

dxxdxx 2

1

Examples1) Find

2) Find

3) Find

Cx

dxx 3

32

Cx

Cx

dxxdxx

3

34

4 3

1

3

1

Cx

dxxdxx 23

2

3

2

1

Examples1) Find

2) Find

3) Find

Cx

dxx 3

32

Cx

Cx

dxxdxx

3

34

4 3

1

3

1

Cx

Cx

dxxdxx 3

2

23

2

3

2

3

2

1

Examples of Common Integrals

1) Find

2) Find

dxxcos

dx

x21

1

Examples of Common Integrals

1) Find

2) Find

dxxcos

dx

x21

1

Cx sin

Examples of Common Integrals

1) Find

2) Find

dxxcos

dx

x21

1

Cx sin

Cx 1sin

Integral Formulas to Memorize

• The same as all of the derivative formulas that are memorized.

• List on pg. 357 (and inside front cover of textbook).

More Difficult Examples

1) Find

2) Find

xdxcos5

dxxx 2

More Difficult Examples

1) Find

2) Find

xdxcos5

dxxx 2

xdxcos5

More Difficult Examples

1) Find

2) Find

xdxcos5

dxxx 2

xdxcos5 Cx )sin(5

More Difficult Examples

1) Find

2) Find

xdxcos5

dxxx 2

xdxcos5 Cx )sin(5 Cx sin5

More Difficult Examples

1) Find

2) Find

xdxcos5

dxxx 2

xdxcos5 Cx )sin(5 Cx sin5

Cxx

32

32

More Examples (S)

3) Find

4) Find

dxxxx 1723 26

dxx

xx

4

42 2

More Examples (S)

3) Find

4) Find

dxxxx 1723 26

dxx

xx

4

42 2

Cxxxx 237

2

7

3

2

7

3

More Examples

3) Find

4) Find

dxxxx 1723 26

dxx

xx

4

42 2

Cxxxx 237

2

7

3

2

7

3

dxx )2( 2

More Examples

3) Find

4) Find

dxxxx 1723 26

dxx

xx

4

42 2

Cxxxx 237

2

7

3

2

7

3

dxx )2( 2

Cxx

21

1

More Examples

3) Find

4) Find

dxxxx 1723 26

dxx

xx

4

42 2

Cxxxx 237

2

7

3

2

7

3

dxx )2( 2

Cxx

21

1

Cxx

21

Last Example5) Find dx

x

x 2sin

cos

Last Example5) Find dx

x

x 2sin

cosdx

xx

x )

sin

1(

sin

cos

Last Example5) Find dx

x

x 2sin

cosdx

xx

x )

sin

1(

sin

cos

dxxx )(csccot

Last Example5) Find dx

x

x 2sin

cosdx

xx

x )

sin

1(

sin

cos

dxxx )(csccot

Cx csc

Homework:

page 363

# 9 – 33 odd